- Here we're going to consider a closed tank
that is partially filled with a fluid.
So let's take a look.
I'll draw a picture.
I'm gonna assume that the tank looks like this.
As you'll see at the end of the problem,
it really doesn't matter
whether it's circular or rectangular.
So this is partially filled with a fluid,
and so we have air pressure that's in the tank,
that's six pounds per inch square psi,
and the depth of this fluid right here is 10 feet,
and what we're gonna look at is comparing the pressure
at the bottom of the tank here in psi,
if the fluid is water or the fluid is glycerin.
So let's start with the governing equation
for pressure variation in a
stationary incompressible fluid.
So pressure at some point in the fluid,
is going to be equal
to the pressure at the top of the fluid,
plus the specific weight
of whatever the fluid is,
times the height of the fluid.
So for both fluids, water, and glycerin, this is
gonna be equal to six psi,
and this height H is
gonna be equal to 10 feet.
So let's first decide whether
we're gonna use inches or feet.
My suggestion is to use feet since both our height
and specific weight are given in those.
Let's start with calculating the air pressure in the tank.
So we know that that's equal to
six pounds per inches squared,
and we want it in feet squared.
So it's 144 inches squared per foot squared,
so this is going to be equal to 864
pounds per feet squared.
So the first thing we're gonna do is we're gonna start
with the water.
So here our pressure is gonna be equal
to this 864 pounds
per feet squared,
and we're gonna add to it,
the specific weight of the water,
and that's given right here,
62.43 pounds per feet cubed.
We multiply that by our 10 feet,
which is that H and we find
that the pressure when it's water,
is equal to 1488.3
pounds per feet squared,
and now in order to turn this back into inches,
this time, we divide it by 144,
and so the pressure when the fluid is water,
is equal to 10.3 psi.
Now we're gonna do the same thing for the glycerin.
So the only things
remember that we're going to change,
well certainly the pressure is gonna change,
this is gonna stay the same,
864 pounds per feet squared,
and now we're going to have the specific weight here
of the glycerin, which is
78.6 pound force
per feet cubed,
and again, we multiply this by the height,
which is 10 feet,
and this ends up being
1,650 pounds
per feet squared,
and again, we divide it by 144,
and so now the pressure
at the bottom of the tank, if the fluid is glycerin,
is equal to 11.4
PSI or pounds per square inch.
Note, we don't need to know the diameter
of the tank or the dimensions of the tank,
only the height,
since pressure changes only in the Z direction.
In addition, since we know the air pressure
and the tank, IE, the pressure above the fluid,
we don't need to know the height of the tank.
But why do we assume that air pressure stays constant
even though when we go back here
this is our air,
and the rest is our fluid?
So there is a difference in height and air.
Well, if you look, the specific weight
of air is equal to
0.07516
pounds per feet cube,
multiplying this number by any reasonable height
we'll give a value that's negligible when it's compared
to our six psi, which we're given
or 864 pounds per feet cubed.
So we can safely assume that the air pressure is constant
in the volume of the tank that is not occupied by the fluid.